A Fuzzy Logic Based Resolution Principal for Approximate Reasoning

In this article, we present a systemic approach toward a fuzzy logic based formalization of an approximate reasoning methodology in a fuzzy resolution, where we derive a truth value of A from both values of B → A and B by some mechanism. For this purpose, we utilize a t-norm fuzzy logic, in which an implication operator is a root of both graduated conjunction and disjunction operators. Furthermore by using an inverse approximate reasoning, we conclude the truth value of A from both values of B → A and B, applying an alto-gether different mechanism. A current research is utilizing an approximate reasoning methodology, which is based on a similarity relation for a fuzzification, while similarity measure is utilized in fuzzy inference mechanism. This approach is applied to both generalized modus-ponens/modus-tollens syllog-isms and is well-illustrated with artificial examples.


Introduction
This study is a continuation of a research, which is based on a proposed t-norm fuzzy logic, presented in [1].Here we also use an automated theorem proving, where a resolution principal is a rule of an inference, leading to a refutation theorem-proving technique.Applying the resolution rule in a suitable way, it is possible to check whether a propositional formula is Universally Valid (UV) and construct a proof of a fact that relative consequent's first-order formula is UV or non UV.In 1965, J. A. Robinson [2] introduced the resolution principle for first-order logic.A fuzzy resolution principal, in its part, was introduced by M. Mukaidono [3].
Taking into account the above mentioned, we present the following.
A fuzzy resolvent of two fuzzy clauses 1 C and 2 C , containing the complementary literals i x and i x ¬ respectively, is defined as ( ) where L , 2 L are fuzzy clauses, which don't contain i x and i x ¬ respectively.The operator ∨ is understood as the dis- junction of the literals present in them.It is also a logical consequence of

C C
∧ .A resolution deduction of a clause C from a set S of clauses is a fi- nite sequence of clauses 1 2 , , , n C C C C = such that each i C is either a member of or is a resolvent of two clauses taken from the resolution principle in propositional logic we deduce that, if S is true under some truth valuation v , then ( ) for all i , and in particular, ( ) Example 1: Here is a derivation of a clause from a set of clauses presented by means of a resolution Tree in Figure 1.
In first order logic, resolution condenses the traditional syllogism of logical inference down to single rule.
A simple resolution scheme is: Consequent: .a The entire historical analysis of this approach toward applying of a resolution principal to a logical inference is presented in [4].

Basic Theoretical Aspects
First, Let us consider that A, A', B and B' are fuzzy concepts represented by fuzzy sets in universe of discourse U, U, V and V, respectively and correspondent fuzzy sets be represented as such Consequent: x is A'.
We shall transform the disjunction form of rule into fuzzy implication from fuzzy logic, introduced in [1], or fuzzy relation and apply the method of inverse approximate reasoning to get the required resolvent.However, in the case of complex set of clauses the method is not suitable.Hence, we investigate for another method of approximate reasoning based on similarity to get the fuzzy resolvent.
Let us consider Generalized Fuzzy Resolution first.The key operation used in this method is disjunction.The disjunction operation ∨ is presented in Table S1 and, being applied to above introduced fuzzy sets A and B, looks like that [1] , 1, 1, 1 Whereas correspondent conjunction operation ∧ is also presented in Table S1 and looks like that [1] , 1, 0, 1 Taking into account (2.5) and (2.6) and the fact that If there are two fuzzy clauses 1 2 , , R C C is a fuzzy resolvent of them with keyword i x , then the following inequality holds ( ) ( ) ( ) where T(x) is a truth value of an x.
Proof: Since whereas from (1.1) ( ) From (2.8) let define the following val- ues of truth: Since 1 i x x + ¬ ≡ , then from (2.9) we are getting that In a meantime from the same (2.8)we have ( ) From (2.11) let's take a note that ( ) Continuing from (2.11) let ( ) And finally from (1.1) we have Let's rewrite (2.8) in the following way ( ) From (2.16) given both (2.10) and (2.12) we have Taking into account (2.12) finally we are getting Furthermore from the same (2.16)let Note that from (2.18) 0 i x = , which means that from ( , 1, 0,

T T T T T T T C C T T T T T T
Taking into account (2.21) from (2.15) and (1.1) ( ) , 0.25 T R C C < , then the following is true: Proof: First note that for [ ] From (1.1) and (2.15), given (2.25) we are getting the following ( ) ( ) First from (2.14) we have the following But from (2.13) we have the following , 0.25 T R C C ≥ , then the following is true: , which means that ( ) From (2.14), (2.17) and (2.19) we are getting ( )    implication operator in a fuzzy logic, used in this article is defined as the following (see Table S1 and Table S2) Let us consider a set of cases. •

∩
, i.e. a fuzzy formula (2.30) is not UV, a logical contradiction takes place.

and and and n
contradictive, but in a meantime ( ) 0.5 and given conditions (2.32) we have , which means that a fuzzy formula (2.30) is not UV or is contradictive.
At last let a fuzzy formula (2.30) be UV and also let if a fuzzy sub formula (antecedent) from (2.30)

and and and n
[ ] Based on these results we formulate the following Theorem 2 If there are two fuzzy clauses 1 2 , C C and ( ) ( )

33) Journal of Software Engineering and Applications
Proof: By Definition 1 and in accordance with (2.5) C C are both UV, i.e. ( ) and (2.36) the following is taking place Taking into account that ( ) ( ) together and get the following , R C C is UV.Therefore by Definition 2 we are getting a fact that if 1 2 , C C are both UV, and then ( ) Let us present some considerations about using a notion of similarity, which plays a fundamental role in theories of knowledge and behavior and has been dealt with extensively in psychology and philosophy.A careful analysis of the different similarity measures reveals that it is impossible to single out one particular similarity measure that works well for all purposes.We will utilize a consistent approach toward definition of a similarity measure, based on the same fuzzy logic we used above [1].But this time we will use the operation Equivalence (see Table S1).
Suppose U be an arbitrary finite set, and ( ) be the collection of all fuzzy subsets of U .For ( ) S A B U or simply ( ) , S A B which can also be considered as a function In order to provide a definition for similarity index, a number of factors must be considered.

Definition 3 A function ( )
, S A B defines a similarity between fuzzy concepts , A B if it satisfies the following axioms: If a function ( ) , S A B is defined as operation equivalence from Table S1, then it could be considered as a similarity measure.

Proof:
From Table S1 we have , , then the following is also true: From (2.38) and (2.42) we have , then the following is also true:

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To illustrate our further research before giving the definition of similarity index, we will present couple examples.
Let A and B be two normal fuzzy sets defined over the same universe of discourse U and presented by unimodal linear monotonic membership func- tions and From (2.46) we are getting . This value perfectly matches our intuition and perception of a closeness of terms "SMALLER THAN LARGE" and "LARGE" and membership functions of correspondent fuzzy sets.
In Table 2 there are six sets of pairs of indices .
. This value is in a middle of a scale [0, 1] and also perfectly matches our intuition and perception of an average closeness of terms "LARGE" and "MEDIUM" and membership functions of correspondent fuzzy sets.
Similarly in  , max , 0.5 . This value is also in a middle of a scale [ ] 0,1 and also perfectly matches our intuition and perception of an average closeness of terms "SMALL" and "MEDIUM" and membership functions of correspondent fuzzy sets.
In Table 4 there are eleven sets of pairs of indices . .This value also perfectly matches our intuition and perception of a fact that terms "SMALL" and "LARGE" has nothing in common.In Table 5 there are twelve sets of pairs of indices .
. This value is a confirmation of a fact that both fuzzy sets are identical.

Generalized Fuzzy Resolution Based Approximate Reasoning
Let us remind that the scheme for Generalized Fuzzy Resolution (2.3) Consequent: x is A'.The classical logic equivalence (3.2) can be extended in fuzzy logic with implication and negation functions.We use the same fuzzy logic, which operations are presented in Table S1.Let us first proof that (3.2) holds.
Since , Whereas given (3.6) a binary relationship for the fuzzy conditional proposition of the type: "If y is B ¬ then x is A" for a fuzzy logic is defined as Given an implication operator from Table S1 expression (3.7) looks like From (3.13) and given subsets from (3.14) we have To illustrate these results we will present couple examples.

Example 1
Let U and V be two universes of discourses and correspondent fuzzy sets are represented as in (3.6) [ ] linguistic scale could consist of the terms like {"SMALL"…, "MEDIUM"…, "LARGE"}.Let us consider the following cases.
A labeled "LARGE" The negation of a fuzzy set B would look like A labeled "LARGE" The negation of a fuzzy set B would look like The binary relationship matrix   In the sequel we will use the following lemma.

Generalized Modus Tollens Based Inverse Approximate Reasoning
Let us remind that the scheme for Generalized Modus Tollens The classical logic equivalence (4.2) can be extended in fuzzy logic with implication and negation functions.We use the same fuzzy logic, which operations are presented in Table S1.Let us first proof that (4.2) holds.
words the following is true.

[
in accordance with Definition 2 a fuzzy concept B is also UV, i.e. ( ) 0.5T B ≥.An

subsets of 2 ℜ
revisit the fuzzy conditional inference rule(3.5).It will be shown that when the membership function of the observation B ¬ is continuous, then the conclusion A depends continuously on the observation; and when the mem- bership function of the relation ( ),R B A ¬ is continuous then the observation A has a continuous membership function.We start with some definitions.A fuzzy set A with membership function called a fuzzy number if A is normal, continuous, and convex.The fuzzy numbers represent the continuous possibility distributions of fuzzy terms of the following type level set of a fuzzy interval A is a non-fuzzy set denoted by [ ] d denotes the classical Hausdorff metric expressed in the family of compact fuzzy sets A and B both have finite support { }
Table 3 there are six sets of pairs of indices
from the same Example 1, which confirms results of both Theorem 4 and Theorem 5.